Try downloading WordPress again, access your server via SFTP or FTP, or a file manager in your hosting account’s control panel, and delete then replace your copies of everything except the wp-config.php file and the /wp-content/ directory with fresh copies from the download. This will effectively replace all of your core files without damaging your content and settings.

Some uploaders tend to be unreliable when overwriting files, so don’t forget to delete the original files before replacing them.

Curl is actually not new to WordPress 4.6, and has been in use for many versions. The Requests class system is new, but curl existed in previous versions of the HTTP API.

So, if you just started getting this message, then it’s likely that your host only recently disabled curl for some reason.

Additionally, the new Requests_Transport_cURL class does indeed check for function_exists('curl_exec') before attempting to use it, so it seems likely that your host is using suhosin or some other means of disabling functions which doesn’t cause PHP to properly report that function as disabled.

cURL is disabled on a lot of servers, and may not be enabled by the end user

also, Samuel
If wordpress uses function_exists, they use it wrongly, as I have run a simple script to test for curl on my server, and it does indeed test properly for curl, and returns that it is not available

Try downloading WordPress again, access your server via SFTP or FTP, or a file manager in your hosting account’s control panel, and delete then replace your copies of everything except the wp-config.php file and the /wp-content/ directory with fresh copies from the download. This will effectively replace all of your core files without damaging your content and settings.

Some uploaders tend to be unreliable when overwriting files, so don’t forget to delete the original files before replacing them.